Difference between revisions of "File:ExpQ2plotT.png"

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[[Explicit plot]] of [[exponential]] to base $b\!=\!\sqrt{2} \approx 1.414213562373095$
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Importing image file
  
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The thick curve: $y=\exp_b(x)$.
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  +
The thin line shows the identical funciton, $y\!=\!x$.
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  +
The [[fixed point]]s $L\!=\!2$ and $L\!=\!4$ are solutions of the equation
  +
  +
: $\exp_b(L)=L$
  +
  +
for $b=\sqrt{2}$.
  +
  +
Any of these fixed points can be used to construct a [[superexponential]] to base $\sqrt{2}~$
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<ref>
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http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
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http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf
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D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
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</ref>.
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==References==
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<references/>
  +
  +
http://en.wikipedia.org/wiki/Square_root_of_2
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  +
http://en.wikipedia.org/wiki/Exponential_function
  +
  +
==C++ generator of curves==
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#include<math.h>
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#include<stdio.h>
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#include<stdlib.h>
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#define DB double
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#define DO(x,y) for(x=0;x<y;x++)
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#include "ado.cin"
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  +
DB B=sqrt(2.);
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  +
main(){ int m,n; double x,y; FILE *o;
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o=fopen("ExpQ2plot.eps","w"); ado(o,1204,804);
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fprintf(o,"602 2 translate 100 100 scale\n");
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#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
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#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
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for(m=-6;m<7;m++) {M(m,0)L(m,8)}
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for(m=0;m<9;m++) {M(-6,m)L(6,m)}
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fprintf(o,"2 setlinecap .01 W S\n 2 setlinecap 1 setlinejoin \n");
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for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W .8 0 0 RGB S\n");
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M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
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fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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system("epstopdf ExpQ2plot.eps");
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system( "open ExpQ2plot.pdf");
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getchar(); system("killall Preview");//for mac
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}
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  +
==[[Latex]] generator of labels==
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% file ExpQ2plot.pdf should be generated with the code above in order to compile the Latex document below.<br>
  +
% Copyleft 2012 by Dmitrii Kouznetsov<br>
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%<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\paperwidth 1212pt
  +
\paperheight 844pt
  +
\topmargin -92pt
  +
\oddsidemargin -80pt
  +
\textwidth 1604pt
  +
\textheight 1604pt
  +
\pagestyle {empty}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
{\begin{picture}(1202,802)
  +
\put(590,792){\sx{4.2}{$y$}}
  +
\put(590,698){\sx{4.2}{$7$}}
  +
\put(590,598){\sx{4.2}{$6$}}
  +
\put(590,498){\sx{4.2}{$5$}}
  +
\put(590,398){\sx{4.2}{$4$}}
  +
\put(590,298){\sx{4.2}{$3$}}
  +
\put(590,198){\sx{4.2}{$2$}}
  +
\put(590,098){\sx{4.2}{$1$}}
  +
\put(080,-22){\sx{4}{$-5$}}
  +
\put(180,-22){\sx{4}{$-4$}}
  +
\put(281,-22){\sx{4}{$-3$}}
  +
\put(381,-22){\sx{4}{$-2$}}
  +
\put(482,-22){\sx{4}{$-\!1$}}
  +
\put(603.6,-22){\sx{4}{$0$}}
  +
\put(703.7,-22){\sx{4}{$1$}}
  +
\put(803.8,-22){\sx{4}{$2$}}
  +
\put(903.9,-22){\sx{4}{$3$}}
  +
\put(1004.0,-22){\sx{4}{$4$}}
  +
\put(1104.1,-22){\sx{4}{$5$}}
  +
\put(1192.2,-22){\sx{4.3}{$x$}}
  +
%\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}}
  +
\put(1034,490){\sx{5.6}{\rot{66}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
  +
\put(1094,444){\sx{5.7}{\rot{45}$y\!=\!x$\ero}}
  +
\put(10,10){\ing{ExpQ2plot}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
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[[Category:Book]]
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[[Category:BookPlot]]
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[[Category:Base sqrt2]]
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[[Category:Exponential]]
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[[Category:Fixed point]]
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[[Category:Transfer function]]
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[[Category:SuperFunctions]]
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[[Category:AbelFunctions]]
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[[Category:Explicit plot]]
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[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:35, 1 December 2018

Explicit plot of exponential to base $b\!=\!\sqrt{2} \approx 1.414213562373095$

The thick curve: $y=\exp_b(x)$.

The thin line shows the identical funciton, $y\!=\!x$.

The fixed points $L\!=\!2$ and $L\!=\!4$ are solutions of the equation

$\exp_b(L)=L$

for $b=\sqrt{2}$.

Any of these fixed points can be used to construct a superexponential to base $\sqrt{2}~$ [1].

References

  1. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
    http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

http://en.wikipedia.org/wiki/Square_root_of_2

http://en.wikipedia.org/wiki/Exponential_function

C++ generator of curves

#include<math.h>
#include<stdio.h>
#include<stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include "ado.cin"

DB B=sqrt(2.);

main(){ int m,n; double x,y; FILE *o;
o=fopen("ExpQ2plot.eps","w"); ado(o,1204,804);
fprintf(o,"602 2 translate 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
for(m=-6;m<7;m++) {M(m,0)L(m,8)}
for(m=0;m<9;m++) {M(-6,m)L(6,m)}
fprintf(o,"2 setlinecap .01 W S\n 2 setlinecap 1 setlinejoin \n");
 for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W .8 0 0 RGB S\n");
M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  system("epstopdf ExpQ2plot.eps");
  system(    "open ExpQ2plot.pdf");            
  getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% file ExpQ2plot.pdf should be generated with the code above in order to compile the Latex document below.
% Copyleft 2012 by Dmitrii Kouznetsov

%


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 1212pt
\paperheight 844pt
\topmargin -92pt
\oddsidemargin -80pt
\textwidth 1604pt
\textheight 1604pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\pagestyle{empty}
\begin{document}
{\begin{picture}(1202,802)
\put(590,792){\sx{4.2}{$y$}}
\put(590,698){\sx{4.2}{$7$}}
\put(590,598){\sx{4.2}{$6$}}
\put(590,498){\sx{4.2}{$5$}}
\put(590,398){\sx{4.2}{$4$}}
\put(590,298){\sx{4.2}{$3$}}
\put(590,198){\sx{4.2}{$2$}}
\put(590,098){\sx{4.2}{$1$}}
\put(080,-22){\sx{4}{$-5$}}
\put(180,-22){\sx{4}{$-4$}}
\put(281,-22){\sx{4}{$-3$}}
\put(381,-22){\sx{4}{$-2$}}
\put(482,-22){\sx{4}{$-\!1$}}
\put(603.6,-22){\sx{4}{$0$}}
\put(703.7,-22){\sx{4}{$1$}}
\put(803.8,-22){\sx{4}{$2$}}
\put(903.9,-22){\sx{4}{$3$}}
\put(1004.0,-22){\sx{4}{$4$}}
\put(1104.1,-22){\sx{4}{$5$}}
\put(1192.2,-22){\sx{4.3}{$x$}}
%\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}}
\put(1034,490){\sx{5.6}{\rot{66}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
\put(1094,444){\sx{5.7}{\rot{45}$y\!=\!x$\ero}}
\put(10,10){\ing{ExpQ2plot}}
\end{picture}}
\end{document}

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